What Is a Zero Sum Game and Why Is It Important in 2019?

You and your friend would both like to eat this M&M. You would each like to eat an entire candy and, even if you could practically split it in half, have no interest in doing so. You stare at each other over this candy. A flock of doves has taken flight. Someone puts on a peppy 60s song.

One of you will get this M&M. One of you will lose it.

You have entered a zero sum game.

What Is a Zero Sum Game?

The zero sum game is an idea from game theory. It finds most of its application in economics and political theory.

In a zero sum game, gains for one person(s) causes losses for another person(s) in an identical amount. The net change for everyone involved is zero and no wealth is created or destroyed during the transaction.

From our candy example above, one of you will end up +1 M&M while another ends up -1 M&M.

Zero Sum Game Characteristics

• Demand that exceeds the supply of a resource,

• A pool of resources that neither expands nor contracts, and

• A net change of zero to all participants.

In this situation, where supply is insufficient to meet demand and cannot be adjusted upward, the only way to distribute a resource to some people is for an equal number of people not to get it.

Odds and Evens Example

The classic example of a zero sum game is odds and evens.

In this game we have two players, Odd and Even. Each of them guesses a number either 1 or 2. If their guess matches, then player Even wins the bet. If their guesses don't player Odd wins the bet. The bet is $1. Here's how their game would play:

Player Even wins: Transfer +1/-1

Player Odd wins: Transfer -1/+1

This is a zero sum game. For Even to win a dollar, Odd has to lose that dollar. The same holds in reverse. The sum of gain and loss in this game is always zero.

The Closed Universe

One of the most important elements of a zero sum game is the closed, inelastic universe.

The players in a zero sum game must start with all the resources they are ever going to have. The supply of resources can't change, it cannot grow or shrink, and there can be no external intervention.

This is because a zero sum game models resource movement and competition only. Under any other conditions a player's gain can outweigh another player's loss (a positive sum game) or vice versa (a negative sum game). Take our candy example again. In our current hypothetical, you and your friend end with a zero sum equilibrium of +1/-1 delicious chocolates.

Now let's consider a few other versions:

• You do not start with the M&M.

You would each like an M&M, but don't start with it. In this case one of you still gets an M&M and the other does not. But because there was no M&M to lose, your net result is +1/0. This is a net gain, not zero sum.

• You each start with an M&M.

Same conditions, but this time both players start with an M&M. In this case each of you ends with an M&M, meaning that the final holding is +1/+1. In either case this is not a zero sum game, since each player can have an M&M without the other player losing one.

• You each start with an M&M but would like two M&Ms.

We're back in a zero sum game. Both players start with an M&M, but both players would like to end with two candies. As a result, the final holding for one player will be 2. Since there are only two candies in the entire transaction the transaction must be +1/-1, netting zero.

Multiplayer Zero Sum Games

You will see that almost all of the examples available online for a zero sum game are two player. This is because, from a mathematical perspective, it becomes exponentially harder to calculate the results of a zero sum game as you add players.

Indeed, mathematically representing the results of multiplayer games is often impossible.

Zero sum games can have any number of players, however, as long as the essential definition remains true: The net gain and loss among all players must be zero.

Take, for example, a small corporate department at bonus time. The manager has $5,000 to split among his five employees. He could spread the money evenly, giving each employee $1,000 apiece, but he feels that some have outperformed others this year. He would like to give out merit-based bonuses.

For the employees this has become a zero sum game. For one worker to get more in the annual bonus other workers must get less. For example, let's say the boss gives one worker $2,000 and two workers $500 each. We get a transfer of +1,000/-500/-500/0/0. The net gain and loss here adds up to zero among all three players.

Zero Sum Games And Economics

In a nutshell, trade and tariffs.

Ultimately, zero sum games are usually an intellectual exercise. In some circumstances the zero sum game accurately describes the real world. Futures contracts, for example, involve a trade where one holder of the contract will pay and the other will collect.

However in almost all trades, deals and financial contracts the world is far bigger than a zero sum game because almost nothing happens in a closed universe. Stock investors, for example, put money into a company. This doesn't become a clean situation where they will either gain or lose, because it's possible for the company to use their investment to gain overall value. It's possible for everyone to get wealthier than when they started.

Or take our bonuses example above. Let's say all five employees decide they want to do better than their $1,000 bonus next year. If they work harder and bring more money into the firm they can expand the available pool of bonus money, creating a potential for net gain.

This is true about trade and finance in general.

In almost all deals, each party gives away something that has less value to them in exchange for something that has more value. In a successful transaction everyone gets wealthier because they have something that increases their current value in some way.

Take, for example, a farmer buying $500 worth of apple seeds. In a closed universe this is a zero sum transaction: the farmer gives the store money and receives an equivalent amount of seeds. In the real world, however, this is potentially a positive sum transaction. The farmer will plant these seeds, and use the resulting fruit to generate wealth. The shop owner will reinvest this money in new products which will also, hopefully, generate wealth.

Each party gets something that has more value to them.

Zero Sum Games And The Trump Tariffs

You will have read a lot about zero sum games recently because this is the basis of the Trump tariffs.

President Donald Trump has often represented foreign trade as a zero sum game, one in which each nation gains and loses in equal measure. Each dollar that America spends buying goods from overseas is a dollar that the country loses. Under this theory trade deficits are unambiguously bad.

This is a fundamentally incorrect view of foreign trade. Per Adam Smith:

It is the maxim of every prudent master of a family, never to attempt to make at home what it will cost him more to make than to buy. What is prudence in the conduct of every private family, can scarce be folly in that of a great kingdom. If a foreign country can supply us with a commodity cheaper than we ourselves can make it, better buy it of them with some part of the produce of our own industry.

Foreign trade, when conducted well, can be positive sum. At its most basic, in a good deal each nation gives something that it creates efficiently for something that it makes inefficiently; or something of which it has a surplus for something that it lacks.

Take a country rich in iron mines. Raw iron for industry is cheap here because the country can produce it in large volumes. Its neighbor has a healthy banking sector and makes a lot of money. As a result, our Iron Nation has a surplus of raw materials while our Banking Country has a lot of capital.

The two countries trade. Rather than run its own iron mines and produce metal expensively, Banking Country buys its raw materials from Iron Nation. It runs a trade deficit in doing so. On the surface this would make our capital-intense nation look like the loser in a zero sum exchange. In reality, however, both nations profit. Banking Country would spend far more producing its own iron, so it ends up with both more money and more iron than it would have had otherwise. Iron Nation gains a market for a product it has in abundance, generating money it hasn't created internally.

Does this mean all trade is good? Certainly not. Just like positive sum deals are real so are negative sum exchanges, in which both players lose. And it's quite possible to have a positive sum deal in which one player gets less value than it could have elsewhere.

However it also means that the world is far, far more complex than simple trade deficits make it seem.